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A058307 a(n) = (n+1)*a(n-1) + a(n-2), with a(0)=0, a(1)=1. 15
0, 1, 3, 13, 68, 421, 3015, 24541, 223884, 2263381, 25121075, 303716281, 3973432728, 55931774473, 842950049823, 13543132571641, 231076203767720, 4172914800390601, 79516457411189139, 1594502063024173381, 33564059780918830140, 740003817243238436461 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Numerator of convergent to BesselI(0,2)/BesselI(1,2) for which the continued fraction expansion is [1,2,3....,n]. - Benoit Cloitre, Mar 27 2003

Numerator of continued fraction C(n) minus denominator of continued fraction C(n), where C(n) = [ 1; 2,3,4,...n ]. - Melvin Peralta, Jan 17 2017

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

C. Cannings, The Stationary Distributions of a Class of Markov Chains, Applied Mathematics, Vol. 4 No. 5, 2013, pp. 769-773.

S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.

Russell Walsmith, Cl-Chemy II

FORMULA

a(2*r + 1) = Sum_{j=0..r} (binomial(r + j, r - j)*(r + j)!/(r - j)! - binomial(r + j, r - j - 1)*(r + j + 1)!/(r - j)!)  and a(*r) = Sum_{j=0..r} (binomial(r + j + 1, r - j](r + j + 1)! /(r - j)! - binomial(r + j, r - j)*(r + j + 1)!/(r - j + 1)! + binomial(r + j + 1, r - j)*(r + j + 1)!/(r - j)!). - Wouter Meeussen, Feb 02 2001

E.g.f.: Pi*(BesselI(2, 2)*BesselY(2, 2*I*sqrt(1-x)) - BesselY(2,2*I)*BesselI(2, 2*sqrt(1-x)))/(1-x). Motivated to look into e.g.f.'s for such recurrences by email exchange with Gary Detlefs. One has to use simplifications after differentiation and putting x=0. See Abramowitz-Stegun handbook p. 360, 9.1.16. - Wolfdieter Lang, May 18 2010

Limit n->infinity a(n)/(n+1)! = BesselI(0,2)-BesselI(1,2) = 0.688948447698738204... - Vaclav Kotesovec, Jan 05 2013

a(n) = Sum_{k = 0..floor((n-1)/2)} (n-2*k-1)!*binomial(n-k-1,k)*binomial(n-k+1,k+2). Cf. A058798. - Peter Bala, Aug 01 2013

a(n) = (n+1)!*hypergeometric([1/2-n/2,1-n/2],[3,-1-n,1-n],4)/2 for n >= 2. - Peter Luschny, Sep 10 2014

MAPLE

A058307 := proc(n) option remember; if n <= 1 then n else A058307(n-2)+(n+1)*A058307(n-1); fi; end;

MATHEMATICA

RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == a[n-2] + a[n-1] (n+1)}, a, {n, 30}] (* Vincenzo Librandi, May 06 2013 *)

Table[FullSimplify[(-BesselI[2+n, -2] * BesselK[2, 2] + BesselI[2, 2] * BesselK[2+n, 2]) / (BesselI[3, 2] * BesselK[2, 2] + BesselI[2, 2] * BesselK[3, 2])], {n, 0, 20}] (* Vaclav Kotesovec, Feb 14 2014 *)

PROG

(MAGMA) [n le 2 select n-1 else Self(n-2)+Self(n-1)*(n): n in [1..30]]; // Vincenzo Librandi, May 06 2013

(Sage)

def A058307(n):

    if n < 2: return n

    return factorial(n+1)*hypergeometric([1/2-n/2, 1-n/2], [3, -1-n, 1-n], 4)/2

[round(A058307(n).n(100)) for n in (0..21)] # Peter Luschny, Sep 10 2014

(PARI) m=30; v=concat([1, 3], vector(m-2)); for(n=3, m, v[n]=(n+1)*v[n-1] +v[n-2]); concat([0], v) \\ G. C. Greubel, Nov 24 2018

(Sage) def A058307(n, D={}):

    if D.has_key(n):

        return D[n]

    else:

        if (n==0): result = 0

        elif (n==1): result = 1

        else: result = expand((n+1)*A058307(n-1) + A058307(n-2))

    D[n] = result

    return result

[A058307(n) for n in range(30)] # G. C. Greubel, Nov 24 2018

CROSSREFS

A column of A058294. Except for first term, -1 times row sums of A053495.

Cf. A001053, A060997, A058798.

Sequence in context: A125279 A186371 A121954 * A020107 A284718 A284719

Adjacent sequences:  A058304 A058305 A058306 * A058308 A058309 A058310

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 09 2000

STATUS

approved

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Last modified June 19 00:55 EDT 2019. Contains 324217 sequences. (Running on oeis4.)